Non-linear Scaling of SRR

Typically, SRR has a unit cell size that is 1/10^th [23] to 1/11^th [13] of the incident wavelength. However, as the operating frequency becomes higher and higher, non-linear scaling has been reported. Many approaches have been applied to study the physical reason and mechanism of this effect.

Ishikawa and T. Tanaka reported the effects of the surface resistivity and the internal reactance on the magnetic responses of the SRR by considering the delay of the current in the conductor [33]. The frequency dependence of the surface resistivity and internal reactance is depicted in Figure 3-1. As the frequency increases, both surface resistivity and internal reactance of the metals increase. The increase of the surface resistivity results in the decrease of the Q value of the SRR, and thereby, degrades the tunable range of the permeability μ. The increase of the internal reactance leads to the reduction of the SRR resonant frequency. The surface resistivity saturates at the inherent frequency of each metal about 10 THz. The saturation value of silver is remarkably smaller than those of gold and copper. The internal reactance, on the other hand, does not saturate and moves away from zero drastically as the frequency increases. As a result, in the optical frequency region, the effect of the internal reactance on the resonant frequency of the SRR was found to be more dominant than that of the surface resistivity. It can be seen in Figure 3-2 that as the geometric size of the SRR decreases, the magnetic response that is defined by the difference between maximum and minimum values of the effective μ decreases. And the resonant frequency shifts to a higher value. Note that the increase of the resonant

frequency is not linearly proportional to the sizes of the SRR due to the increase of the internal reactance.

Figure 3-1 Dispersion curves of the internal impedance of silver, gold, and copper.

In the frequency region exceeding THz, the internal reactance is more dominant than the surface resistivity, and this internal reactance decreases the resonant frequency.

The figure is imaged from [33].

Figure 3-2 Real and imaginary parts of the effective permeability of the silver SRRs as a function of the SRR pattern size. The labeling in each case indicates the unit-cell size a. The inset shows the corresponding sizes of the SRR in nanometers. The figure is imaged from [33].

Figure 3-3 Frequency dependencies of the minimum value of Re(μ) of the SRRs made of (a) silver, (b) gold, and (c) copper, with the filling factor of F = 5%, 7%, 10%, and 13%, respectively. The frequency dependencies of the imaginary part of the effective permeability lRe whose real part has the minimum value are also shown.

Only the silver SRR exhibits negative l in the visible range of 400–700 THz. The figure is imaged from [33].

The frequency dependence of the minimum value of Re(μ) of the SRR for three kinds of metals and four different filling factors were plotted in Figure 3-3. As the frequency increases, the minimum value of Re(μ) approaches unity asymptotically.

Only the silver SRR exhibits negative Re(μ) in the visible light range, while the propagation loss of silver is higher than that of gold or copper. Note that the filling factor is also important for the realization of the negative μ in visible light range.

Further studies showed that above the linear scaling regime the resonance frequency saturates due to the free electron kinetic energy, which cannot be neglected any longer in comparison with the magnetic energy [34]. Although the presence of the magnetic resonance was reported to be observed with very small unit cells, such a weak resonance was unable to conduct negative values as Re(μ) approaches unity. The maximum resonance frequency (and thus the negative Re(μ) region) increases with the number of cuts in the SRR. The 4-cut SRR retains the negative Re(μ) region for higher frequencies up to about 550 THz. As a comparison, the 2- and 1-cut SRR could only reach up to 420 THz and 280 THz (unit cell size ~60 nm), respectively. The geometries of the 1-, 2-, and 4-cut single-ring SRR are schematically presented in Figure 3-4 (a). Note that the gap widths were designed to make the capacitance of the three types of SRR approximately the same, but the experimental result revealed that the three capacitances were indeed different. This is because the formula

d C ~ wt is

not valid when d is not much smaller than w. Moreover, the periodic boundary condition along the direction of the electric field only adds asymmetric side capacitances to the SRR, as which is represented by the equivalent circuit in Figure 3-4 (b). Therefore, the three types of SRR have inequivalent capacitances and thus different saturation frequencies. The scaling of the magnetic resonance frequency as a function of the unit cell size a is shown in Figure 3-5.

The electron kinetic energy adds an electron self-inductance Le to the magnetic response which scales as 1/a while both the magnetic field inductance Lm and capacitance C of the SRR scale as a. The resonance frequency fm thus has the following a-dependence:

where c1 and c2 are independent of a. Other factors that break the linear scaling and contribute to the increase of the Ohmic losses are (1) the increased scattering of electrons at the surface of the metal and (2) the larger skin depth (scales as

a

1 ) over

metal thickness ratio. Both factors depend in a complicated way on the geometry and the surface smoothness, and thus require further experimental studies.

Figure 3-4 (a) The geometries of the 1-, 2-, and 4-cut single-ring SRR. The SRR is made of aluminum, simulated using Drude model (fp=3570 THz, fτ=19.4 THz). The parameters of the SRR are side length l=0.914a, width and thickness w=t=0.257a, and cut width d=0.2a, 0.1a, and 0.05a for the 1-, 2-, and 4-cut SRR, respectively. (b) The left panel shows the charge accumulation in a 4-cut SRR, as a result of the periodic boundary conditions in the E (and H) direction. The right panel shows the equivalent LC circuit describing this SRR. Cg is the gap capacitance and Cs the side capacitance resulting from the interaction with the neighboring SRR. The figure is imaged from [34].

Figure 3-5 Solid lines with symbols shows the scaling of the simulated magnetic resonance frequency fm as a function of the size of the unit cell a for the 1-, 2-, and 4-cut SRR, respectively. Up to the lower THz region, the scaling is linear, fm∝ 1/a.

The maximum attainable frequency is strongly enhanced with the number of cuts in the SRR ring. The hollow symbols as well as the vertical line at 1/a =17.9 μm^-1 indicate that no μ<0 is reached anymore. The nonsolid lines show the scaling of fm

calculated from Eq. 3.1 (LC circuit model). The figure is imaged from [34].

Despite the small line width that is theoretically required when scaling down the SRR, there are studies showing that the ratio of the effective sizes of the SRR to the wavelength of the incident beam is around 1, which suggests a wide scale independence [35]. This size tolerance is useful in realizing the metamaterials with negative n in the optical wavelength range.

In most experiments, the transmission is measured at a fixed incident angle for a certain range of wavelength. However, in [35], the incident angle was tuned while the wavelength was fixed by choosing different laser sources. Lasers with the output wavelength of 10.86 μm and 9.43 μm were used because the characteristic size of the SRR (10.48μm) was in between these two wavelengths. The transmission results were shown in Figure 3-6 (a) and (b). At 90° tile angle (in-plane incident) the transmission was expected to peak since the direct contribution of the incident beam to the detector

became dominant, as can be seen in the case of Al-coated planar Si (gray line).

However, when the SRRs were patterned on the substrate, absorption at in-plane incident occurred (black line). Although the absorption of parallel polarization at 9.43 μm is less significant than that at 10.86 μm, it is still observable, showing a great size tolerance. The coherence of the laser may be critical to the onset of resonance.

Moreover, an 808 nm laser was used to cross-check the rapid oscillation near the non-resonance region caused by the Fabry-Perot effect, as shown in Figure 3-6 (c).

Here, neither the rapid oscillation nor the magnetic response appeared. The transmission through Si at wavelength below 1μm is extremely low due to high loss.

And the incident wavelength is beyond size tolerance.

Figure 3-6 Normalized transmission of the SRRs. Data of the Al-coated planar Si is presented for reference. The operating wavelength is (a) 10.86 μm, (b) 9.43 μm and (c) 808 nm, and the laser power on the sample is ~100mW. The figure is imaged from [35] and edited